Posterior Distribution Research Articles

When Pleas Precede Evidence

In most U.S. jurisdictions, prosecutors are not required to clearly establish a reasonable basis for guilt prior to offering defendants plea deals. We apply Bayesian analyses, which are uniquely suited to illuminate the impact of prior probability of guilt on the informativeness of a particular outcome (i.e., a guilty plea), to demonstrate the risks of plea offers that precede evidence. Our primary prediction was that lower prior probabilities of guilt would coincide with a significantly higher risk­­ for false guilty pleas. We incorporated data from Wilford, Sutherland et al. (2021) into a Bayesian analysis allowing us to model the expected diagnosticity of plea acceptance across the full range of prior probability of guilt. Our analysis indicated that, as predicted, when plea offers are accepted at lower prior probabilities of guilt, the probability that a plea is actually false is significantly higher than when prior probabilities of guilt are higher. In other words, there is a trade-off between prior probability of guilt and information gain. For instance, in our analysis, when prior probability of guilt was 50%, posterior probability of guilt (after a plea) was 77.8%; when prior probability of guilt was 80%, posterior probability of guilt was 93.3%. Our results clearly indicate the importance of ensuring that there is a reasonable basis for guilt before a plea deal is extended. In the absence of shared discovery, no such reasonable basis can be established. Further, these results illustrate the additional insights gained from applying a Bayesian approach to plea-decision contexts.

Projection predictive variable selection for discrete response families with finite support

AbstractThe projection predictive variable selection is a decision-theoretically justified Bayesian variable selection approach achieving an outstanding trade-off between predictive performance and sparsity. Its projection problem is not easy to solve in general because it is based on the Kullback–Leibler divergence from a restricted posterior predictive distribution of the so-called reference model to the parameter-conditional predictive distribution of a candidate model. Previous work showed how this projection problem can be solved for response families employed in generalized linear models and how an approximate latent-space approach can be used for many other response families. Here, we present an exact projection method for all response families with discrete and finite support, called the augmented-data projection. A simulation study for an ordinal response family shows that the proposed method performs better than or similarly to the previously proposed approximate latent-space projection. The cost of the slightly better performance of the augmented-data projection is a substantial increase in runtime. Thus, if the augmented-data projection’s runtime is too high, we recommend the latent projection in the early phase of the model-building workflow and the augmented-data projection for final results. The ordinal response family from our simulation study is supported by both projection methods, but we also include a real-world cancer subtyping example with a nominal response family, a case that is not supported by the latent projection.

A modified EM-type algorithm to estimate semi-parametric mixtures of non-parametric regressions

Semi-parametric Gaussian mixtures of non-parametric regressions (SPGMNRs) are a flexible extension of Gaussian mixtures of linear regressions (GMLRs). The model assumes that the component regression functions (CRFs) are non-parametric functions of the covariate(s) whereas the component mixing proportions and variances are constants. Unfortunately, the model cannot be reliably estimated using traditional methods. A local-likelihood approach for estimating the CRFs requires that we maximize a set of local-likelihood functions. Using the Expectation-Maximization (EM) algorithm to separately maximize each local-likelihood function may lead to label-switching. This is because the posterior probabilities calculated at the local E-step are not guaranteed to be aligned. The consequence of this label-switching is wiggly and non-smooth estimates of the CRFs. In this paper, we propose a unified approach to address label-switching and obtain sensible estimates. The proposed approach has two stages. In the first stage, we propose a model-based approach to address the label-switching problem. We first note that each local-likelihood function is a likelihood function of a Gaussian mixture model (GMM). Next, we reformulate the SPGMNRs model as a mixture of these GMMs. Lastly, using a modified version of the Expectation Conditional Maximization (ECM) algorithm, we estimate the mixture of GMMs. In addition, using the mixing weights of the local GMMs, we can automatically choose the local points where local-likelihood estimation takes place. In the second stage, we propose one-step backfitting estimates of the parametric and non-parametric terms. The effectiveness of the proposed approach is demonstrated on simulated data and real data analysis.

A statistical analysis method for significant wave height and spectral peak frequency considering the random and time-varying effects based on copula function and Bayesian inference

Random and time-varying effects are important factors for statistical analysis of wave characteristic variables, including the significant wave height and spectral peak frequency. This paper proposes a statistical analysis method for the accurate statistical analysis of the state of the ocean. Several common distributions are applied as candidates for describing a specific variable, denoted as the marginal distribution. The joint distribution for the wave characteristic variables is constructed using copula functions based on the marginal distributions. The probability and unknown parameters of the marginal distributions are then determined by fully Bayesian inference. The best-fitting marginal distribution is selected based on the posterior probabilities of the candidates. Then, unknown parameters of the candidate copula functions are estimated by maximum likelihood estimation. The best-fitting copula function is selected based on Akaike information criterion, root mean squared error and Nash Sutcliffe efficiency. The proposed method is verified using the National Data Buoy Center dataset for 2019. However, this dataset, collected from a network of almost 100 moored buoys and Coastal-Marine Automated Network (CMAN) stations, contains incomplete data. The results reveal that the best-fitting marginal distribution and copula function may vary with the month. The average and maximum values of the improved RMSE using the proposed method are only 0.0064 and 0.0187, respectively. This indicates the high accuracy of the proposed method for the statistical analysis of wave states even though missing some data.

Identifying potential drug targets for non-alcoholic fatty liver disease: a drug target Mendelian randomization study

Abstract Funding Acknowledgements Type of funding sources: Public grant(s) – National budget only. Main funding source(s): UK Research and Innovation Training Grant: Centre for Doctoral Training in AI-enabled Healthcare Systems Background Non-alcoholic fatty liver disease (NAFLD), a range of conditions caused by the build-up of fat in the liver, is the most common form of chronic liver disease, with an estimated prevalence of ~25% globally. NAFLD is associated with increased risk of all-cause mortality, with CVD emerging as the primary cause of death amongst affected individuals. The aetiology of NAFLD is not yet clearly understood, and currently no drugs exist for its treatment, with therapeutics aimed at symptom management instead. Purpose To identify and prioritise candidate protein drug targets and pathways relevant for therapeutic intervention in NAFLD. Methods To identify plasma proteins causally associated with NAFLD, we analysed genetic associations on 1,561 proteins from 35,559 participants in the deCODE study, and compared these to associations from a GWAS of biopsy confirmed NAFLD cases (1,483 cases, 17,781 controls). We sourced genetic variants from a 400kbp region around each protein-coding gene, pruning variants on an F-statistic of 15, minor allele frequency threshold of 1% and linkage disequilibrium r-squared of 0.4. cis-Mendelian randomization (MR) was used to identify proteins associated with NAFLD. Bias due to pleiotropy was limited by excluding variants with leverage statistic more than 3 times the mean, or outlier statistic larger than 10.83. We used colocalization analysis to identify proteins with a posterior probability of at least 0.8, indicating the presence of common causal variants. We further prioritised proteins by annotating them with information on druggability (i.e. whether a protein is a target for a known drug, or predicted to be a target for a drug), mRNA expression in pathology-relevant tissues, and involvement in known biological pathways. Results We identified 5 proteins with a multiplicity-corrected (p<3.2×10-5) association with NAFLD, and strong evidence of genetic colocalization: CYB5A, NT5C, NCAN, TGFBI, DAPK2. TGFBI and DAPK2 are potentially druggable, with all five found to be expressed in both liver and adipose tissues. Pathway analysis indicated the involvement of NT5C, CYB5A, NCAN in Reactome pathway R-HSA-1430728 reflecting cellular energy metabolism, including mitochondrial lipid metabolism, which is strongly implicated with NAFLD. Conclusion We identified a prioritised set of five plasma proteins influencing NAFLD, including druggable proteins informing de novo and ongoing drug development programs. Figure: MR effect estimates of proteins on NAFLD A: Effect of each protein on NAFLD. Each point represents a protein, effect estimates are reported in log(odds ratio) and statistical significance in -log(p-value). Coloured points represent proteins passing the multiplicity-corrected p-value threshold B: MR effect estimates of prioritised proteins. Estimates are reported in log(odds ratio). Proteins are annotated according to their druggability based on the British National Formulary and ChEMBL

Songbird annual productivity declines with increasing soil lead contamination around nests

Abstract Songbirds incidentally ingest soil contaminated with lead and several species in the Southeast Missouri Lead Mining District have a negative relationship between soil lead concentration and reproduction. We used an individual-based model (IBM) to simulate nesting processes throughout the breeding season to estimate annual productivity for 5 songbirds in relation to soil-lead concentration. We modeled daily nest survival and number fledged in relation to soil lead and incorporated these relationships into the IBM using a Bayesian approach that fully captured parameter uncertainty and process variation. The proportion of the posterior distribution for the effect of soil lead on daily nest survival was mostly negative for Pipilo erythrophthalmus (Eastern Towhee), Passerina cyanea (Indigo Bunting), Cardinalis cardinalis (Northern Cardinal), Sialia sialis (Eastern Bluebird; f = 0.972, 0.990, 0.741, 0.581, respectively), and slightly positive for Spizella pusilla (Field Sparrow; f = 0.680). The proportion of the posterior distribution for the effect of soil lead on the number fledged from successful nests was mostly negative for S. pusilla, P. erythrophthalmus, and C. cardinalis (f = 0.867, 0.585, 0.508, respectively) and positive for P. cyanea and S. sialis (f = 0.773, 0.744). Simulated annual productivity decreased by 0.04–1.47 young/female/year among species across the range of soil lead concentrations, with the greatest declines for P. erythrophthalmus and P. cyanea. The probability of fledging one or more young changed from 51% to 15% for P. erythrophthalmus, 57% to 23% for P. cyanea, 60% to 48% for C. cardinalis, 94% to 84% for S. sialis, and 57% to 64% for S. pusilla as soil-lead concentration around a nest increased from 20 to 4,000 ppm. Most nest failures were the result of predation; therefore, we suggest lead may have affected birds’ behavior at the nest, or affected nest site quality, which resulted in higher nest predation.

A Computationally Efficient Approach to Fully Bayesian Benchmarking

In small area estimation, it is sometimes necessary to use model-based methods to produce estimates in areas with little or no data. In official statistics, we often require that aggregates of small area estimates agree with national estimates for internal consistency purposes. Enforcing this agreement is referred to as benchmarking, and while methods currently exist to perform benchmarking, few are ideal for applications with non-normal outcomes and benchmarks with uncertainty. Fully Bayesian benchmarking is a theoretically appealing approach insofar as we can obtain posterior distributions conditional on a benchmarking constraint. However, existing implementations may be computationally prohibitive. In this paper, we critically review benchmarking methods in the context of small area estimation in low- and middle-income countries with binary outcomes and uncertain benchmarks, and propose a novel approach in which posterior samples of small area characteristics from an unbenchmarked model can be combined with a rejection sampler or Metropolis-Hastings algorithm to produce benchmarked posterior distributions in a computationally efficient way. To illustrate the flexibility and efficiency of our approach, we provide comparisons to an existing benchmarking approach in a simulation, and applications to HIV prevalence and under-5 mortality estimation. Code implementing our methodology is available in the R package stbench.

Optimal Joint Defense and Monitoring for Networks Security under Uncertainty: A POMDP-Based Approach

The increasing interconnectivity in our infrastructure poses a significant security challenge, with external threats having the potential to penetrate and propagate throughout the network. Bayesian attack graphs have proven to be effective in capturing the propagation of attacks in complex interconnected networks. However, most existing security approaches fail to systematically account for the limitation of resources and uncertainty arising from the complexity of attacks and possible undetected compromises. To address these challenges, this paper proposes a partially observable Markov decision process (POMDP) model for network security under uncertainty. The POMDP model accounts for uncertainty in monitoring and defense processes, as well as the probabilistic attack propagation. This paper develops two security policies based on the optimal stationary defense policy for the underlying POMDP state process (i.e., a network with known compromises): the estimation-based policy that performs the defense actions corresponding to the optimal minimum mean square error state estimation and the distribution-based policy that utilizes the posterior distribution of network compromises to make defense decisions. Optimal monitoring policies are designed to specifically support each of the defense policies, allowing dynamic allocation of monitoring resources to capture network vulnerabilities/compromises. The performance of the proposed policies is examined in terms of robustness, accuracy, and uncertainty using various numerical experiments.

Dengue fever mapping in Bangladesh: A spatial modeling approach

AbstractBackgroundEpidemics of the dengue virus can trigger widespread morbidity and mortality along with no specific treatment. Examining the spatial autocorrelation and variability of dengue prevalence throughout Bangladesh's 64 districts was the focus of this study.MethodsThe spatial autocorrelation is evaluated with the help of Moran and Geary . Local Moran was used to detect hotspots and cold spots, whereas local Getis Ord G was used to identify only spatial hotspots. The spatial heterogeneity has been detected using various conventional and spatial models, including the Poisson‐Gamma model, the Poisson‐Lognormal Model, the Conditional Autoregressive (CAR) model, the Convolution model, and the BYM2 model, respectively. These models are implemented using Gibbs sampling and other Bayesian hierarchical approaches to analyze the posterior distribution effectively, enabling inference within a Bayesian context.ResultsThe study's findings show that Moran and Geary analysis provides a substantial clustering pattern of positive spatial autocorrelation of dengue fever (DF) rates between surrounding districts at a 90% confidence interval. The Local Indicators of Spatial Autocorrelation cluster mapped spatial clusters and outliers based on prevalence rates, while the local Getis‐Ord G displayed a thorough breakdown of high or low rates, omitting outliers. Although Chattogram had the most dengue cases (15,752), Khulna district had a higher prevalence rate (133.636) than Chattogram (104.796). The BYM2 model, determined to be well‐fitted based on the lowest Deviance Information Criterion value (527.340), explains a significant association between spatial heterogeneity and prevalence rates.ConclusionThis research pinpoints the district with the highest prevalence rate for dengue and the neighboring districts that also have high risk, allowing government agencies and communities to take the necessary precautions to mollify the risk effect of DF.

Probabilistic Prediction of Floating Offshore Wind Turbine Platform Motions via Uncertainty Quantification and Information Integration

The accurate prediction of short-term platform motions in a real environment is crucial for the safe design, operation, and maintenance of floating offshore wind turbines (FOWTs). Numerical simulations of motions are typically associated with high uncertainties due to abstracted theoretical models, empirical parameters, initial environment parameters, etc. Therefore, it is necessary to integrate other sources of information associated with less uncertainty, e.g., monitoring data, for accurate predictions. In this paper, we propose a probabilistic prediction based on the Bayesian approach that logically integrates motion monitoring data with simulated motion predictions of FOWTs, considering uncertainties in the environment model, structural properties, motion prediction method, monitoring data, etc. The approach consists of constructing a prior probability density function (PDF) of a random variable (which characterizes the largest value of the initial motion response) via numerical simulations and a likelihood function based on platform motion monitoring data and deriving a posterior PDF of the random variable by Bayesian updating. Then, posterior distributions of short-term extreme motion responses are derived using the posterior PDF of the random variable, representing lower uncertainty and improved accuracy. A Metropolis–Hastings algorithm is adopted to obtain PDFs of complex probability distributions. The effectiveness of the approach is demonstrated on a real FOWT platform in Scotland. The proposed probabilistic prediction approach results in posterior distributions of short-term extreme platform motions associated with less uncertainty and higher accuracy, which is attributed to integrating prior knowledge with monitoring data.

Bayesian analysis of physiologically based toxicokinetic (PBTK) modeling for pentachlorophenol exposure in pregnant women

Pentachlorophenol (PCP) is a persistent organic compound that is widely present in the environment. The estimation of internal exposure levels for a given external exposure using toxicokinetic models is key to the human health risk assessment of PCP. The present study developed a physiologically based multicompartmental pharmacokinetic (PBTK) model to describe and predict the behavior of pentachlorophenol (PCP) in an organism. The model consists of stomach, intestines, adipose tissue, kidneys and fast- and poorly perfused tissues that are interconnected via blood circulation. We constructed a PBTK model of PCP in rats and extrapolated it to human dietary PCP exposure. The toxicokinetic data of PCP in human tissues and excreta were obtained from the published literature. Based on the collected PCP dietary survey and internal exposure data of pregnant women in Shanghai, Bayesian statistical analysis was performed for the model using Markov chain Monte Carlo (MCMC) simulation. The posterior distributions of the sensitive parameters were estimated, and the model was parameter optimized and validated using the pregnant women's test dataset. The results showed that the root mean square error (RMSE) improved 37.3% compared to the original model, and a systematic literature search revealed that the optimized model achieved acceptable prediction results for other datasets in China. A PCP metabolism model based on the exposure characteristics of pregnant women in China was constructed in the present study. The model provides a theoretical basis for the study of PCP toxicity and risk assessment.

THE UNINFORMATIVE PRIOR OF JEFFREYS’ DISTRIBUTION IN BAYESIAN GEOGRAPHICALLY WEIGHTED REGRESSION

When using the Bayesian method for estimating parameters in a geographically weighted regression model, the choice of the prior distribution directly impacts the posterior distribution. The distribution known as the Jeffreys prior is an uninformative type of prior distribution and is invariant to reparameterization. In cases where information about the parameter is not available, the Jeffreys' prior is utilized. The data was fitted with an uninformative Jeffreys' prior distribution, which yielded a posterior distribution that was utilized for estimating parameters. This study aims to derive the prior and marginal posterior distributions of the Jeffreys' and in Bayesian geographically weighted regression (BGWR). The marginal posterior distributions of and can be obtained by integrating the other parameters of a common posterior distribution. Based on the results and discussion, the Jeffreys prior in BGWR with the likelihood function is . On the other hand, the marginal posterior distribution of follows a normal multivariate distribution, that is, , while the marginal posterior distribution of follows an inverse gamma distribution, that is, . As further research, it is necessary to follow up on several limitations of the results of this research, namely numerical simulations and application to a particular case that related to the results of the analytical studies that we have carried out.

Decision-making in the multiphase optimization strategy: Applying decision analysis for intervention value efficiency to optimize an information leaflet to promote key antecedents of medication adherence.

Advances in the multiphase optimization strategy (MOST) have suggested a new approach, decision analysis for intervention value efficiency (DAIVE), for selecting an optimized intervention based on the results of a factorial optimization trial. The new approach opens possibilities to select optimized interventions based on multiple valued outcomes. We applied DAIVE to identify an optimized information leaflet intended to support eventual adherence to adjuvant endocrine therapy for women with breast cancer. We used empirical performance data for five candidate leaflet components on three hypothesized antecedents of adherence: beliefs about the medication, objective knowledge about AET, and satisfaction with medication information. Using data from a 25 factorial trial (n = 1603), we applied the following steps: (i) We used Bayesian factorial analysis of variance to estimate main and interaction effects for the five factors on the three outcomes. (ii) We used posterior distributions for main and interaction effects to estimate expected outcomes for each leaflet version (32 total). (iii) We scaled and combined outcomes using a linear value function with predetermined weights indicating the relative importance of outcomes. (iv) We identified the leaflet that maximized the value function as the optimized leaflet, and we systematically varied outcome weights to explore robustness. The optimized leaflet included two candidate components, side-effects, and patient input, set to their higher levels. Selection was generally robust to weight variations consistent with the initial preferences for three outcomes. DAIVE enables selection of optimized interventions with the best-expected performance on multiple outcomes.

Synergizing habits and goals with variational Bayes

Behaving efficiently and flexibly is crucial for biological and artificial embodied agents. Behavior is generally classified into two types: habitual (fast but inflexible), and goal-directed (flexible but slow). While these two types of behaviors are typically considered to be managed by two distinct systems in the brain, recent studies have revealed a more sophisticated interplay between them. We introduce a theoretical framework using variational Bayesian theory, incorporating a Bayesian intention variable. Habitual behavior depends on the prior distribution of intention, computed from sensory context without goal-specification. In contrast, goal-directed behavior relies on the goal-conditioned posterior distribution of intention, inferred through variational free energy minimization. Assuming that an agent behaves using a synergized intention, our simulations in vision-based sensorimotor tasks explain the key properties of their interaction as observed in experiments. Our work suggests a fresh perspective on the neural mechanisms of habits and goals, shedding light on future research in decision making.

Combining prior and post‐release data while accounting for dispersal to improve predictions for reintroduction populations

AbstractAttempts to reintroduce species to managed areas may be compromised by dispersal into the surrounding landscape. Therefore, decisions regarding the selection and ongoing management of reintroduction areas require predicting dispersal as well as the survival and reproduction rates of the species to be reintroduced. Dispersal can potentially be measured directly by tracking animals, but this is often impractical. However, dispersal can also be inferred from re‐sighting surveys done within reintroduction areas if such data are available from multiple areas with varying connectivity to the surrounding landscape, allowing apparent survival and recruitment to be modelled as a function of connectivity metrics. Here, we show how data from 10 previous reintroductions of a New Zealand passerine, the toutouwai (Petroica longipes), were used to predict population dynamics at a predator‐controlled reintroduction area with high connectivity, and predictions then updated using post‐release data. Bayesian hierarchical modelling of the previous data produced prior distributions for productivity, adult survival and apparent juvenile survival rates that accounted for random variation among areas as well as rat density and connectivity. The modelling of apparent juvenile survival as a function of connectivity allowed it to be partitioned into estimates of survival and fidelity. Bayesian updating based on post‐release data produced posterior distributions for parameters that were consistent with the priors but much more precise. The prior data also allowed the recruitment rate estimated in the new area to be partitioned into separate estimates for productivity, juvenile survival and juvenile fidelity. Consequently, it was possible to not only estimate population growth under current management, but also predict the consequences of reducing the scale or intensity of predator control, facilitating adaptive management. The updated model could then be used to predict population growth as a function of the connectivity and predator control regime at proposed reintroduction areas while accounting for random variation among areas.

Proteome-wide Mendelian randomization identifies potential therapeutic targets for nonalcoholic fatty liver diseases

Nonalcoholic fatty liver disease (NAFLD) is the predominant cause of liver pathology. Current evidence highlights plasma proteins as potential therapeutic targets. However, their mechanistic roles in NAFLD remain unclear. This study investigated the involvement of specific plasma proteins and intermediate risk factors in NAFLD progression. Two-sample Mendelian randomization (MR) analysis was conducted to examine the association between plasma proteins and NAFLD. Colocalization analysis determined the shared causal variants between the identified proteins and NAFLD. The MR analysis was applied separately to proteins, risk factors, and NAFLD. Mediator shares were computed by detecting the correlations among these elements. Phenome-wide association studies (phewas) were utilized to assess the safety implications of targeting these proteins. Among 1,834 cis-protein quantitative trait loci (cis-pQTLs), after-FDR correction revealed correlations between the plasma levels of four gene-predicted proteins (CSPG3, CILP2, Apo-E, and GCKR) and NAFLD. Colocalization analysis indicated shared causal variants for CSPG3 and GCKR in NAFLD (posterior probability > 0.8). Out of the 22 risk factors screened for MR analysis, only 8 showed associations with NAFLD (p ≤ 0.05), while 4 linked to CSPG3 and GCKR. The mediator shares for these associations were calculated separately. Additionally, reverse MR analysis was performed on the pQTLs, risk factors, and NAFLD, which exhibited a causal relationship with forward MR analysis. Finally, phewas summarized the potential side effects of associated-targeting proteins, including CSPG3 and GCKR. Our research emphasized the potential therapeutic targets for NAFLD and provided modifiable risk factors for preventing NAFLD.

Sequential Monte Carlo with Adaptive Lookahead Support for Improved Importance Sampling

The sequential Monte Carlo, also called the Bayesian particle filter, approximates a posterior probability density function of a latent target state from noisy sensor measurements using a set of Monte Carlo samples. These samples are predicted using an importance density function and then updated using the Bayes’s rule. The updated samples and their corresponding weights provide an estimate of the latent state. The said filtering process is iterated over time for tracking dynamic target states. It is critical to have enough particles in regions of the target state space that contribute to the posterior. The auxiliary and the improved auxiliary particle filters accomplish this by a process that mimics drawing from an importance density that leverages the incoming observation into the sampling step. However these filters are known to fail when the sensor measurements are highly informative and the diffusion over the state transition is large. This paper presents an improvement to the auxiliary particle filter by taking two support points that act as limits in a univariate state space within which particles are samples. The choice of the limits is adaptive. The proposed method is successfully tested using a nonlinear model using simulations.

Enabling identification of component processes in perceptual learning with nonparametric hierarchical Bayesian modeling.

Perceptual learning is a multifaceted process, encompassing general learning, between-session forgetting or consolidation, and within-session fast relearning and deterioration. The learning curve constructed from threshold estimates in blocks or sessions, based on tens or hundreds of trials, may obscure component processes; high temporal resolution is necessary. We developed two nonparametric inference procedures: a Bayesian inference procedure (BIP) to estimate the posterior distribution of contrast threshold in each learning block for each learner independently and a hierarchical Bayesian model (HBM) that computes the joint posterior distribution of contrast threshold across all learning blocks at the population, subject, and test levels via the covariance of contrast thresholds across blocks. We applied the procedures to the data from two studies that investigated the interaction between feedback and training accuracy in Gabor orientation identification over 1920 trials across six sessions and estimated learning curve with block sizes L = 10, 20, 40, 80, 160, and 320 trials. The HBM generated significantly better fits to the data, smaller standard deviations, and more precise estimates, compared to the BIP across all block sizes. In addition, the HBM generated unbiased estimates, whereas the BIP only generated unbiased estimates with large block sizes but exhibited increased bias with small block sizes. With L = 10, 20, and 40, we were able to consistently identify general learning, between-session forgetting, and rapid relearning and adaptation within sessions. The nonparametric HBM provides a general framework for fine-grained assessment of the learning curve and enables identification of component processes in perceptual learning.

Bayesian analysis of the molecular emission and dust continuum of protoplanetary disks

The MIRI instrument on board the James Webb Space Telescope probes the chemistry and dust mineralogy of the inner regions of protoplanetary disks. The observed spectra are unprecedented in their detail and reveal a rich chemistry with strong diversity between objects. This complicates interpretations that are mainly based on manual continuum subtraction and 0D slab models. We investigate the physical conditions under which the gas emits in protoplanetary disks. Based on MIRI spectra, we apply a full Bayesian analysis that provides the posterior distributions of dust and molecular properties, such as column densities and emission temperatures. To do so, we introduced the Dust Continuum Kit with Line emission from Gas (DuCKLinG), a Python-based model simultaneously describing the molecular line emission and the dust continuum of protoplanetary disks without large computational cost. The model describes the dust continuum emission by dust models with precomputed dust opacities. The molecular emission is based on LTE slab models but from extended radial ranges with gradients in column densities and emission temperatures. We compare the model to observations using Bayesian analysis with linear regression techniques to reduce the dimension of the parameter space. We benchmarked this model to a complex thermo-chemical ProDiMo model of AA\,Tau and fit the MIRI spectrum of GW\,Lup. The latter allowed for a comparison to the previous results obtained with single slab models and hand-fitted continuum. We successfully decrease the computational time of the fitting method by a factor of $80$ by eliminating linear parameters, such as the emission areas, from the Bayesian run. This approach does not significantly change the retrieved molecular parameters, and only the calculated errors on the optically thin dust masses slightly decrease. For an AA\,Tau ProDiMo mock observation, we find that the retrieved molecular conditions from DuCKLinG (column densities from $3 cm^ $ to $4 cm^ $, radial range from $0.2\ au $ to $1.2\ au $, and temperature range from about $200\ K $ to $400\ K $) fall within the true values from ProDiMo (column densities between $4 cm^ $ to $5 cm^ $, radial extent $0.1\ au $ to $6.6\ au $, and temperature range from about $120\ K $ to $1000\ K $). The smaller DuCKLinG ranges can be explained by the relative flux contributions of the different parts of ProDiMo. The parameter posterior of GW\,Lup reinforces previously found results. The previously determined column densities fall within the retrieved ranges in this study for all examined molecules ( CO2 H2O HCN and C2H2 ). Similar overlap is found for the temperatures with only the temperature range of HCN (from $570^ K$ to K$) not including the previously found value ($875\ K$). This discrepancy may be due to the simultaneous fitting of all molecules compared to the step-by-step fitting of the previous study. There is statistically significant evidence for radial temperature and column density gradients for H2O and CO2 compared to the constant temperature and column density assumed in the 0D slab models. Additionally HCN and C2H2 emit from a small region with near constant conditions. Due to the small selected wavelength range $13.6\ m$ to $16.3\ m$, the dust properties are not well constrained for GW\,Lup. DuCKLinG can become an important tool to analyse the molecular emission and dust mineralogy of large samples based on JWST/MIRI spectra in an automated way.

Stochastic acceleration in extreme TeV BL Lacs through MCMC

Extreme TeV BL Lacs are a class of blazars with unique spectral and temporal features that are not easily reproducible using standard one-zone models based on single shock acceleration. To account for their peculiar properties, we elaborated a two-step acceleration model in which a recollimation shock and the subsequent downstream turbulence energize non-thermal electrons. We applied the model to a sample of extreme TeV BL Lacs with well-characterized spectral energy distributions. Since we used several sources, we automatized the exploration of the parameter space. This allowed us to derive the parameter distributions and study the correlations among them. We numerically solved a system of two coupled nonlinear differential equations to obtain the non-thermal particles and turbulence spectra. We calculated the spectral energy distribution via the synchrotron self-Compton emission model. The automatization of the parameter space exploration is possible through a Markov chain Monte Carlo (MCMC) ensemble sampler, in our case emcee We derived well-defined posterior distributions for the parameters, showing that the model is well constrained by available data and demonstrating the suitability of our method. The cross-correlations among some of the physical parameters are not trivial.\ Therefore, we conclude that MCMC sampling is a key instrument for characterizing the complexity of our multiparameter phenomenological model.